Beware of mathematicians!

yesterday part

Now please take a pen and draw a straight line that goes through points "1" and "2". Measure the size between your dots and draw the point "0" before "1". It would be nice, if the lengths from "0" to "1" and from "1" to "2" are equal, however this is not important.

Imagine two small mathematical demons standing at the point "0". They are big enough to go forward on top of your line. The black demon holds a flag with "{i : i ∈ N}" on it. He counts natural numbers. The red demon holds a flag with "{2*i : i ∈ N}". He counts only even natural numbers.

They start to go forward and they both make one move each second. The black "i" steps on the point "1", the red "2*i" jumps on the point "2". Then the "i" steps on the point "2". The "2*i" jumps again two times longer.

A mathematics teacher would draw a big circle and label it with "N" then a smaller one within it and label the smaller with something like "N₂". The sentence you would hear during this may sound like "The set of even natural numbers is included in the set of natural numbers".

The representation is obvious, the definition is clear. Does this mean that there are two times more natural numbers than even natural numbers? (We use the definition of "N" that excludes the number "0".)
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Beware of mathematicians!

yesterday part

The last exercise does not add anything new to the understanding of the nature of mathematics but it is important for further description of the confrontation between mathematicians and computers.

Let's take the line segment between dots "0" and "1" and summon the tabletop rational demon. This is the green demon who has a magical ruler that can exactly measure the distance between two mathematical points and a magical cutter that cuts a line segment at exact mathematical point. The width of the cut is exactly 0 in any measurement units but you can imagine that the demon leaves a thin green mark on the paper surface in the place of the cut.

The rational demon sets the exact positions for points "0" and "1", measures with his ruler the distance between them and cuts this segment into two halves. He has produced the number "1/2". We usually write it as "0.5".

Now he divides distance between "0" and "1" into three exactly equal segments and cuts the numbers "1/3" and "2/3".

Let's stop and look at the number "1/3". This is a perfectly simple rational number. But if we try to use the decimal representation, we get a strange construction "0.33333333333333…"

Ellipsis "…" means that this sequence is infinite.

By the way, behind "0.5" also lies an infinite sequence. Unfortunately mathematics teachers rarely mention this.
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